Answer
$w = 68^{\circ}$
$x = 151.6$
Work Step by Step
Since we have the measure of an angle and the measure of the hypotenuse, we can use the sine ratio to find $w$, which is the side opposite to the given angle.
The sine ratio is given as follows:
sin $A = \frac{opposite}{hypotenuse}$
Let's plug in what we know:
sin $42^{\circ} = \frac{w}{102}$
Multiply each side by $102$:
$102(sin 42^{\circ}) = w$
Multiply to solve for $w$:
$w = 68^{\circ}$
With the other right triangle, we are given the measure of the angle and the hypotenuse. We are asked to find half of the adjacent side marked $x$, which is the adjacent side of this particular right triangle. Let's use the cosine ratio:
cos $A = \frac{adjacent}{hypotenuse}$
Let's plug in what we know:
cos $42^{\circ}$ = $\frac{x/2}{102}$
Multiply each side by $102$:
(cos $42^{\circ}$)$(102) = x/2$
Multiply each side by $2$:
$x = 2 (cos $42^{\circ}$)$(102)$
Evaluate to solve for $x$:
$x = 151.6$
